and zero for the rest of the indices. In particular, for the acoustic crystal from Fig. REF , we had \(\mathbf {a}_i = \theta \mathbf {a}^{\prime }_i\) , hence \(\theta _{13}=-\theta _{31}=\theta _{24} = - \theta _{42} = \theta \) . This is also the case for the pattern c from Fig. REF , but the \(\Theta \) -matrix for the pattern b from Fig. REF  contain more entries. As a consequence, the non-commutative 4-torus is not just a simple product of two non-commutative 2-tori, as it was the case in the previous experimental works.[1]}, [2]}
